I would go with the coconut oil since a common cause is the fungus Malassezia. The caprylic acid of coconut oil is an effective antifungal. In addition coconut oil is a surfactant that will help to cut the excess oil production.
After washing the hair it would also be a good idea to apply a citric acid solution to the hair and scalp to restore the natural acidity. I have been experimenting with different acid concentrations and am currently using a mixture of citric and malic acid, 1/2 teaspoon each per 4 ounces purified water. Been pretty happy with it. citric acid is easier to find, and you can just use one teaspoon per 4 ounces water.
It would also be a good idea to address this internally. Digestive bitters can help balance out the hormones that can also contribute to overproduction of oils.